Page 70 - Schaum's Outline of Differential Equations

CHAP. 7] APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 53

Fig. 7-3 Fig. 7-4

ORTHOGONAL TRAJECTORIES
Consider a one-parameter family of curves in the xy-plane defined by

where c denotes the parameter. The problem is to find another one-parameter family of curves, called the
orthogonal trajectories of the family (7.72) and given analytically by

such that every curve in this new family (7.13) intersects at right angles every curve in the original family (7.72).
We first implicitly differentiate (7.72) with respect to x, then eliminate c between this derived equation and
(7.72). This gives an equation connecting x, y, and y', which we solve for / to obtain a differential equation of
the form

The orthogonal trajectories of (7.72) are the solutions of

(See Problems 7.23-7.25.)
For many families of curves, one cannot explicitly solve for dyldx and obtain a differential equation of the
form (7.14). We do not consider such curves in this book.

Solved Problems

7.1. A person places $20,000 in a savings account which pays 5 percent interest per annum, compounded
continuously. Find (a) the amount in the account after three years, and (b) the time required for the
account to double in value, presuming no withdrawals and no additional deposits.
Let N(t) denote the balance in the account at any time t. Initially, N(Q) = 20,000. The balance in the account
grows by the accumulated interest payments, which are proportional to the amount of money in the account. The
constant of proportionality is the interest rate. In this case, k= 0.05 and Eq. (7.1) becomes

This differential equation is both linear and separable. Its solution is